The current Swedish Royal Family comes from the House of Bernadotte, which was founded in 1810.
Jean Bernadotte, a Frenchman, was one of Napoleon's top Generals. Because the Swedish King had no heir, the Riksdag of the Estates, as the Swedish Parliament was known, started scouring around for a replacement. Usually they would have invited the second son of an ally's King to their throne, to cement relations between the countries, but in the early 19th century, Napoleon was the top dog in Europe, conquering country after country, and installing his brothers as kings in some of the minor countries.
In order to avoid being conquered themselves, and to have a king that Napoleon approved of, they elected Bernadotte in 1810 to be King. That way they could be independent (because the new King wasn't a blood relative of Napoleon's) but also kind of friendly (because he was French and a favoured General). The new king called himself King Charles XIV John of Sweden and Napoleon was said to have been surprised at the appointment.
A few years later, Napoleon's fortunes started to turn. The British, along with their allies in Portugal and Prussia were closing on him. In 1813 Bernadotte and his Parliament decided to pivot, break relations with the French and back the British. It paid off handsomely in 1815, when the British defeated Napoleon at Waterloo. All the states that Napoleon had conquered were returned to their former status, and his brothers removed as kings.
But Bernadotte continued as King of Sweden, and his descendants are still on the Swedish throne. In addition, one of his descendants is the current King of Norway. And that is how the monarchy of the Scandinavian states got their French flavour!